Monopoly pricing with dual capacity constraints Robert Somogyi



JOB MARKET PAPER September 14, 2015

Abstract This paper studies the price-setting behavior of a monopoly facing two capacity constraints: one on the number of consumers it can serve, the other on the total amount of products it can sell. Facing two consumer groups that differ in their demands and the distribution of their willingness-to-pay, the monopoly’s optimal non-linear pricing strategy consists of offering one or two price-quantity bundles. The characterization of the firm’s optimal pricing in the short run as a function of its two capacities reveals a rich structure that also gives rise to some surprising results. In particular, I show that prices are non-monotonic in capacity levels. Moreover, there always exists a range of parameters in which weakening one of the capacity constraints decreases consumer surplus. In the long run, when the firm can choose how much capacity to build, prices and consumer surplus are monotonic in capacity costs.

JEL Classification: D21, D42, L12 Keywords: Capacity constraint, Monopoly pricing, Multi-dimensional capacity



Ecole Polytechnique and CREST. Email address: [email protected] The latest version of this paper is available here.

1

Introduction

The economics literature typically considers one-dimensional capacity constraints (Kreps and Scheinkman [1983]). This is idealized because in real-world production processes firms typically face several capacity constraints (size of plants, inventories, workforce, etc.). In general, due to the use of supply chains, firms are constrained on even more aspects of their production. The objective of this paper is to develop a theory of monopoly pricing in the presence of multiple capacity constraints. Examples of industries characterized by dual capacity constraints range from hospitals, through restaurants, to the freight transport industry. Hospitals are constrained by the number of beds available in their intensive care unit on the one hand, and operating room time on the other hand. Each consumer (i.e. patient) needs one bed but consumers differ in their need of operating room time. Hospitals’ patients have price-inelastic individual demands; even if a long surgery (e.g., a kidney transplant) is very cheap, someone in need of a short surgery (e.g., fixing of a broken arm) will never prefer having the longer one. Restaurants constitute another prominent example for the co-existence of the two types of capacity constraints. Restaurants have to take into account in their pricing decisions both the number of tables they have at their disposal and the size of their kitchen, that limits the amount of food they can prepare. Patrons of restaurants arrive in groups of different sizes, hence they are typically heterogeneous in their consumption and also in their willingness-to-pay. Dual capacities are key characteristics of the freight transport industry as well. Both in ocean container shipping and air cargo transport, an important concern of the transporting company is optimizing the mix of items according to both their size and their weight. As physical and regulatory limits are present in both dimensions, profit-maximizing firms cannot avoid taking into account both constraints. Another class of examples includes markets where firms are constrained by some physical capacity constraint on the one hand and their workers’ time on the other. For instance airplanes cannot fly at full capacity if the ratio of passengers to flight attendants exceeds a certain regulatory limit. Business class passengers tend to use up more of the flight attendants’ time than economy class passengers and the willingness-to-pay of the two groups are obviously different. Several questions arise if one wants to understand the consequences of the co-existence of the two types of capacity constraints. How much will the predictions

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of the model change compared to a model with only one capacity constraint? What are the optimal prices a firm must charge to different consumer groups? How will these optimal prices change as a function of the capacity levels? Under which conditions are both capacity constraints binding in optimum? How do aggregate consumer surplus, profit and total welfare vary with the capacity levels? In order to answer these questions, I consider a price-setting monopolist facing two types of capacity constraints. The firm is unable to serve more than K consumers. In addition, it cannot sell more than a quantity Q of its products. Consumers differ in their price-inelastic demands and their willingness-to-pay (WTP). There are two consumer groups: high-types intend to buy a larger amount of the product than the low-types. The monopoly can observe the demand of each consumer but not their WTP. The WTP of high-types and low-types are distributed along two different intervals. Some high-types have a larger WTP than all the low-types while some low-types have a larger per-unit WTP than all the high-types. The monopoly is allowed to offer different prices for the two quantities in order to discriminate between consumers. The existence of a second type of capacity constraint fundamentally changes the monopoly’s optimal behavior. The optimal pricing strategy in the short run (with exogenously given capacities) is the following. When K, the constraint on the number of consumers, is very tight, the monopoly excludes low-types and serves only high-type consumers. Conversely, when the capacity constraint on total production, Q, is very small then only low-types are served. For larger levels of capacities, it is optimal for the monopoly to serve both types. When both constraints are very large, the firm chooses the unconstrained optimal prices. When K is of intermediate value while Q is very large, optimal prices are chosen in a way that K binds and Q does not. Conversely, if Q is of medium value and K is large, then Q is binding and K is slack. Importantly, there exist capacity levels for which the two prices are chosen so that both constraints bind. Intuitively, when only very few consumers can be served, the monopoly will prefer serving those with the highest overall WTP so it excludes all the low-types. Conversely, when the total production is very limited, the firm is concerned by the per-unit surplus it can extract from the consumers, hence it excludes all the high-types. Even when capacities do not take extreme values and some consumers of both types are served, an increase in K ceteris paribus makes the low-types more attractive to the firm therefore it chooses its prices to attract more low-types. In sharp contrast to the standard case with only one capacity constraint,

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prices are non-monotonic in the size of capacities. In particular, while prices are decreasing in both capacities in most parameter regions, for capacity levels when both constraints bind, the price charged for high-types increases in K while the price for low-types increases in Q. The intuition for this result is the following. For very small values of K, the capacity on the number of people served, the monopoly only serves high-types. As K increases low-types become relatively more valuable for the firm so after a threshold value it starts serving both consumer groups. Below the threshold value the firm decreases the price charged for the high-types so that their number equals K. Above the threshold, in order to make space for the low-types, the firm is interested in serving less high-types so it raises the price charged to them. A similar argument can be made to understand the price increase for the low-types as Q increases. Furthermore, this price increase has a non-negligible effect on aggregate consumer surplus: I show that there always exists a range of parameters in which weakening one of the capacity constraints decreases aggregate consumer surplus. This phenomenon cannot occur in models that consider a single capacity constraint. The decrease of consumer surplus is a consequence of the monopoly adjusting its optimal mix of consumers in the following sense. When K is very small, only high-types get served. As it increases above a threshold value, the firm starts serving some low-type consumers as well. However, as the transition is smooth, close to the turning point the firm still serves many high-types and only a few low-types. The price increase suffered by the numerous high-types dominates the gain of the few low-types that start being served and the aggregate consumer surplus decreases. I also investigate long-run behavior of the monopoly where in addition to prices, it can also choose endogenously how much capacity to build in each dimension. For any positive cost function of capacity building, the firm chooses capacity levels and prices so that both capacities bind. These are exactly the capacity levels for which the prices are increasing and aggregate consumer surplus is decreasing in the short run. I provide a complete characterization of optimal capacity levels for the case of linear cost functions. The outcome of the model gets close to the unconstrained optimum as both costs tend to zero. The optimal consumer mix depends on the relation of the capacity costs to the marginal benefit of serving an additional consumer of a given type. Therefore, depending on parameter values, both low-types and high-types can be excluded in optimum, or some consumers of both groups can be served. Both optimal prices and consumer surplus are

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monotonic in the capacity costs. I also consider an extension which allows low-types to buy a large quantity of the product and freely dispose of the amount they do not consume. Such a model suits better some of the industries cited as examples, for instance the restaurants. I show that this opportunity does not alter any of the results obtained in the more restrictive basic model. Intuitively, consumers’ possibility of free disposal may limit the monopoly’s choice. In case it wants to sell the two quantities at different prices, it must charge a lower price for the smaller quantity, otherwise all consumers would buy the larger bundle for the lower price which is obviously unprofitable for the firm. However, the optimal prices of the basic model satisfy this condition, thus optimal firm behavior is not altered by free disposal. Next, I investigate incentive compatibility for the high-type consumers. In this variant of the model, the high-types are allowed to buy several small bundles to satisfy their demand and I assume that the good is perfectly divisible. A monopoly that wants to sell both types of bundles must choose lower per-unit prices for the large bundle than for the small one, otherwise it can sell exclusively small bundles. I show that the monopoly’s optimal prices are not affected by the perfect divisibility and high-types’ ability to purchase the small bundle. Finally, I provide sufficient conditions on the distribution of consumers such that all the qualitative insights of the model with uniform distribution hold. In particular, I find that if both distributions satisfy the monotone hazard rate condition, then the region where both capacity constraints bind has a well-defined shape; moreover, prices are non-monotonic in capacity levels and I prove the existence of a region where consumer surplus is decreasing in a capacity level. Furthermore, I identify hazard rate dominance relations as sufficient conditions for the exclusion of a consumer group.

1.1

Related literature

The monopoly’s problem of third-degree price discrimination with a single capacity constraint is a textbook exercise (see Besanko and Braeutigam, 2010, p. 507, Exercise 12.6). Therefore, the literature of capacity-constrained pricing has mainly focused on the case of competition. The seminal paper of Kreps and Scheinkman (1983) shows that under certain assumptions, the outcome of a two-stage game where firms first choose capacities then engage in price competition coincides with the Cournot outcome. Davidson and Deneckere (1986) were the first to point out that this result is not robust to the choice of rationing rule. In particular, they show that the results are more competitive than the Cournot outcome for virtually 4

any rationing rule other than the efficient one. Cripps and Ireland (1988) and Acemoglu et al. (2009) both consider capacityconstrained competition when firms face consumers with price-inelastic demands. They show the existence of pure-strategy subgame perfect equilibria for any capacity levels, which are supported by mixed-strategy equilibria off-path, similarly to Kreps and Scheinkman (1983). Acemoglu et al. (2009) also analyze efficiency properties of the equilibria, i.e., they compare the total social welfare in different equilibria to the welfare-maximizing first-best and they find that some equilibria can be arbitrarily inefficient. Reynolds and Wilson (2000) introduce demand uncertainty into the two-stage game described by Kreps and Scheinkman (1983). They show that symmetric purestrategy equilibria in the capacity choice game do not exist when the variability of demand is high, and they provide a set of assumptions that guarantee the existence of asymmetric pure-strategy equilibria. De Frutos and Fabra (2011) investigate the interaction of demand uncertainty and capacity constraints assuming price-inelastic demand. They identify submodularity of the demand distribution as a sufficient condition of the existence of pure-strategy equilibria and they show that these equilibria are asymmetric. The welfare analysis of the model with general distribution function is related to another stream of recent literature which investigates the welfare effects of a monopoly’s third-degree price discrimination. Cowan (2007) derives two alternative sufficient conditions for the convexity of the slope of demand under which price discrimination enhances social welfare. Aguirre et al. (2010) analyze more general demand functions and identify sufficient conditions on the elasticities and demand curvatures of the two markets that make price discrimination reduce or increase welfare. Cowan (2012) shows how the the cost pass-through coefficient can determine the way discrimination effects aggregate consumer surplus. Finally, Bergemann et al. (2015) investigate welfare effects of additional information a monopoly can use to price discriminate and they show that any combination of consumer and producer surpluses is achievable that satisfies some mild conditions. Models where several capacity constraints co-exist have so far been relegated to the realms of operations research and revenue management. Patient admission planning for scheduled surgeries and patient mix optimization are both important problems hospitals have to face. The multidimensional nature of capacities in hospitals is crucial for such planning. Many recent papers in the operations research literature focus on solving a variety of problems that arise in a context where the

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treatment of different categories of patients require different levels of capacities. For example, Adan and Vissers (2002) take into account operating room time, intensive care unit beds, medium care unit beds and nurses’ time to simulate the optimal schedule of a real-world hospital department. Banditori and al. (2013) also consider a multiple capacity setting for their simulation, in addition, they provide a comparison of the recent articles in this area (Banditori and al., 2013, Table 2). Multidimensional capacities are crucial in freight transport, such as the air cargo industry or the container shipping industry. Since dynamic pricing is widespread in these industries, the literature models freight transport pricing by extending standard revenue management models to accommodate multiple capacities. Xiao and Yang (2010) consider revenue management with two capacity dimensions. Their focus is on ocean container shipping, where container sizes are standardized (two varieties dominate the market) and maximal weights of all containers are set by on-road regulations. They derive an analytical solution and show that under some conditions the optimal policy is qualitatively different when considering the second capacity constraint. Kasilingam (1997) describes how air cargo revenue management is different from air passenger revenue management, and one of the key differences he identifies is the multidimensional aspect of capacities: volume, weight and even cargo position may be constraining. Finally, the hospitality industry’s capacity management literature has also recognized the importance of dual capacities. Kimes and Thompson (2004) optimize the table mix for restaurant revenue management taking into account not only the number and distribution of seats but also the size of the service areas. Bertsimas and Romy (2003) consider both sizes of parties to be seated and expected service duration to compare several optimization-based approaches to restaurant revenue management. The rest of the paper is organized as follows. Section 2 discusses a simple benchmark, then outlines the main model. Section 3 describes the results of the main model, the monopoly’s optimal behavior, and provides comparative statics for the capacity levels. Section 4 discusses the consequences of optimal pricing for consumer surplus. Section 5 generalizes the model by allowing capacity levels to be chosen endogenously. Section 6 investigates a variant of the baseline model with incentivecompatibility. Section 7 analyzes the model with general distribution of consumers and Section 8 concludes. All omitted proofs are relegated to the Appendix.

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2

The model

2.1

A simple benchmark

In this section I present a simple model of monopoly characterized by one capacity constraint facing only one consumer group. Consider a price-setting monopolist that can produce for zero cost up to quantity Q of a good, then his costs become infinite, i.e., the monopoly is characterized by capacity constraint Q. All consumers have a unit demand. Consumers are heterogeneous in their willingness-to-pay (WTP), they are uniformly distributed on the interval [0, 1]. The monopoly can only observe the distribution but not the individual WTP values. The net consumer surplus of a buyer with WTP level w ∈ [0, 1] is given by w − p if he buys the product where p denotes its price, and it is 0 otherwise. A consumer is willing to buy if and only if his net consumer surplus is positive. In this setting, the mass of consumers willing to buy the product at a given price p ∈ [0, 1] is 1−p: this is the fraction of consumers with a higher WTP than the price. Since each consumer has unit demand, the total demand for the product is also 1−p. Hence the monopoly solves the following maximization problem: max π = (1 − p)p

s.t.

p

1−p≤Q

The profit-maximizing price is given by p∗ =

( 1 − Q if Q ≤ 1/2, 1/2

otherwise.

First notice that the capacity constraint is binding up to a certain level (Q ≤ 1/2) then the monopoly can implement its unconstrained optimal strategy. The price is decreasing in the level of capacity, Q, as long as the capacity is small enough to bind, then the price becomes independent of it. Given these prices, total demand is simply Q if Q ≤ 1/2; while for larger values of the capacity it is equal to 1/2. Intuitively, the monopoly chooses prices such that the capacity binds when it is small, then it implements its unconditional optimum. Profit is given by π∗ =

( (1 − Q)Q if Q ≤ 1/2, 1/4

otherwise.

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As one should expect, both the demand and the profit are increasing in capacity for binding levels of capacity. Finally, consumer surplus in this setting is given by (1 − p)2 /2. Clearly, consumer surplus is always decreasing in the price (p never exceeds 1). Hence the fact that the optimal price is decreasing in Q means that consumer surplus is increasing in the level of capacity. This is not surprising: An increase in the level of capacity means more of the consumers are served for a lower price. The four main insights of this simple model are that an increase in the capacity level Q can decrease prices and can increase demand, consumer surplus and profits.1 Out of these 4 insights only the one concerning profits will remain true for a model with dual capacity constraints presented in the next section.

2.2

The dual capacity model

Consider a price-setting monopolist serving a market that consists of two consumers groups. Each consumer is characterized by its demand and its total willingness-topay (WTP) for the product. Low-types want to consume a fix amount of qL > 0 products while high-types want to consume a fix amount of qH > qL , i.e., individual demand is price-inelastic. A consumer of type i ∈ {L, H} with total WTP w has a net consumer surplus of w − pi if he buys a quantity qi of the product for price pi , and 0 otherwise.2 Total WTP of consumers of type i is uniformly distributed on the interval [0, vi ]. Consumers maximize their net surplus and they demand the good if and only if their net surplus is positive. Assume that the total mass of high-type and low-type consumers is αvH and (1 − α)vL , respectively, where 0 ≤ α ≤ 1 scales the relative weight of the two consumer groups.3 The monopoly can observe α, the consumers’ individual demand, and the distribution of their WTP but not their individual values.

Assumption 1. Let the WTP of consumers satisfy the following conditions: 0 < vL < vH

and vL /qL > vH /qH

1

The same insights remain true in case there are 2 consumer groups, as discussed in Section 3.2, “case of very large Q”. 2 In Section 6 I show that all results remain unchanged if one allows low-types to buy the large bundle; or high-types to buy several small bundles. 3 This normalization of the mass of consumers is made to simplify the exposition of results. In particular, it shortens significantly the formulas obtained for optimal prices and profits without altering the qualitative properties of the model.

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Assumption 1 guarantees that some high-type consumers’ valuation always exceed all the low-types’ valuation, whereas the per-unit-WTP of some low-type consumers is greater then the per-unit-WTP of all high-type consumers. This assumption restricts the analysis to the most interesting cases, because otherwise the monopoly would always prefer to serve consumers of one group first, irrespective of the size of capacity constraints. Also, this corresponds to decreasing marginal value of consumption in the present setting. Finally, notice that high-types are not necessarily more valuable for the firm, it simply refers to the high level of their demand. I analyze a monopoly facing two types of capacity constraints: • K denotes the maximal number of consumers the firm can serve • Q denotes the maximal total production of the firm Both constraints are exogenously given. For simplicity, production is costless up to capacity then it becomes impossible. The monopoly has an optimal pricing structure that consist of offering at most 2 price-quantity bundles. Given the price-inelastic demands, no consumer would buy any bundle that offers them a quantity different from their desired demand. Moreover, if the firm were to offer several bundles with the same quantity for a different price, consumers would only buy the cheapest one. Let pH and pL denote the price of the bundle with high and low quantities, respectively.

3

Results

3.1

Optimal monopoly pricing

In this section I describe and solve the monopoly’s profit maximization problem. For any price pH ∈ [0, vH ], the high-type consumers willing to buy are the ones H who have a higher WTP than pH . They represent a fraction vHv−p of the high-types, H which means that the total mass of high-type consumers who demand the good is given by α(vH − pH ). Similarly, for any price pL ∈ [0, vL ], the total mass of low-types willing to buy is (1 − α)(vL − pL ).

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Hence the total demand the monopoly faces at such prices is given by α(vH − pH )qH + (1 − α)(vL − pL )qL The monopoly also has the option of not serving one of the consumer groups. Hence, it must choose between serving both consumer groups, excluding low-type consumers, or excluding high-type consumers. Notice that in some cases the latter possibility can be profitable since some low-type consumers have a higher per-unit WTP than all the high-types. I will break down the general optimization problem into 3 separate maximization problems and compare the locally optimal profits to find the monopoly’s profit-maximizing strategy. The maximization problem of the firm if it decides to exclude low-types writes as

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(P-EL)

max π = α(vH − pH )pH

s.t.

pH

α(vH − pH ) ≤ K

(1a)

α(vH − pH )qH ≤ Q

(1b)

pH ≥ 0

(1c)

The maximization problem if the monopoly serves only low-types is given by

(P-EH)

max π = (1 − α)(vL − pL )pL pL

s.t.

(1 − α)(vL − pL ) ≤ K

(2a)

(1 − α)(vL − pL )qL ≤ Q

(2b)

pL ≥ 0

(2c)

The maximization problem of the firm when serving some consumers of both groups writes as 4 Throughout the paper, EL stands for “excluding low-types” and EH stands for “excluding high-types”.

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(P-LH)

max

pL ,pH

π = α(vH − pH )pH + (1 − α)(vL − pL )pL

s.t.

α(vH − pH ) + (1 − α)(vL − pL ) ≤ K

(3a)

α(vH − pH )qH + (1 − α)(vL − pL )qL ≤ Q

(3b)

pL ≥ 0 , pH ≥ 0

(3c)

pL < vL , pH < vH

(3d)

In each case, the firm maximizes the product of the mass of consumers buying and the price. Constraints 1a, 2a and 3a constitute the upper bound on the maximal number of people that the monopoly can serve, whereas 1b, 2b and 3b are the capacity constraints on total production: The mass of people buying times their demand cannot exceed Q. Constraints 1c, 2c and 3c guarantee non-negativity of the prices. Finally, 3d ensures that some consumers of both types are served in the last case. 3.1.1

Excluding one group of consumers

In problem (P-EL) when the monopoly excludes low types, first notice that the 2 capacity constraints 1a and 1b can be rewritten as α(vH − pH ) ≤ min(K, Q/qH ) Obviously, the unconstrained maximum is attained at pH = vH /2. Thus, the 2 /4 if min(K, Q/qH ) > αvH /2. Otherwise the firm sells up to the profit equals αvH tighter capacity constraint for a price pH = vH − min(K, Q/qH )/α, consequently its profit equals to (min(K, Q/qH ))2 α The non-negativity constraint is trivially satisfied. The solution of the second maximization problem, (P-EH), when the firm excludes high-types, is analogous. Hence the optimal price is given by vH min(K, Q/qH ) −

pL =

( vL − vL /2

min(K,Q/qL ) 1−α

if min(K, Q/qL ) ≤ (1 − α)vL /2, otherwise.

The optimal profit is

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π=

3.1.2

( vL min(K, Q/qL ) −

(min(K,Q/qL ))2 1−α

(1 − α)vL2 /4

if min(K, Q/qL ) ≤ (1 − α)vL /2, otherwise.

Serving both consumer groups

Solving the remaining maximization problem, (P-LH) is more complex and requires writing the Karush-Kuhn-Tucker conditions. I omit the constraints 3c and 3d on prices and show ex-post that the solution of the relaxed problem satisfies them. Let λ1 denote the multiplier of constraint 3a and let λ2 denote the multiplier of 3b. The objective function is

L(pL , pH , λ1 , λ2 ) = α(vH − pH )pH + (1 − α)(vL − pL )pL − λ1 [α(vH − pH ) + (1 − α)(vL − pL ) − K] − λ2 [α(vH − pH )qH + (1 − α)(vL − pL )qL − Q] The first order conditions can be written as 2pH = vH + λ1 + λ2 qH

and

2pL = vL + λ1 + λ2 qL There are 4 cases depending on the sign of the multipliers, i.e., depending on which of the two constraints is binding. The following notation simplifies the exposition of results: let E(x) denote the weighted average of any 2 variables xL and xH , i.e., E(x) = αxH + (1 − α)xL . Case 1 When the capacity constraint on the mass of people served is binding while the other constraint is slack (λ1 > 0 and λ2 = 0), the optimal pricing strategy of the monopoly is given by vH − vL −K 2 and its optimal profit is pL = v L + α

and pH = vH − (1 − α)

 πK ≡ α(1 − α)

vH − vL 2

2

vH − vL −K 2

+ KE(v) − K 2

Primal and dual feasibility of this local optimum require that K, the capacity constraint on the mass of people served, be of an intermediate size with respect to 12

the other parameters of the model: α (vH − vL ) ≤ K ≤ min 2 where

1 g(Q) = E(q)



 E(v) , g(Q) 2

  vH − vL Q − α(1 − α)(qH − qL ) . 2

The non-negativity constraints are satisfied at this solution. Case 2 When the capacity constraint on the total production is binding while the other constraint is slack (λ1 = 0 and λ2 > 0), the optimal pricing strategy of the monopoly is given by   qL α qH pL = (vL + vH )qH + (1 − α)vL qL − Q E(q 2 ) 2 qL and qH pH = E(q 2 )



 1 α qH qH (vL + vH )qH + (1 − α)vL qL − Q − (vL − vH ) 2 qL 2 qL

Its optimal profit is

 2 ! qH − vL + qL    α 1 α qH qH + vH )qH + (1 − α)vL qL − Q Q + (vL − vH )qH + (vL E(q 2 ) 2 qL 2 qL

α πQ ≡ 4

2 vH

Primal and dual feasibility of this local optimum require that Q, the capacity constraint on the total production, be of an intermediate size with respect to the other parameters of the model:   qL E(vq) 1−α qL vL − vH ≤Q≤ and K ≥ f (Q) 2 qH 2 where f (Q) =

E(q) 1 Q + α(1 − α)(vL qH − vH qL )(qH − qL ) E(q 2 ) 2E(q 2 )

Case 3 When both capacity constraints are binding (λ1 > 0 and λ2 > 0), the optimal pricing strategy of the monopoly is given by

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pL = vL −

KqH − Q (1 − α)(qH − qL )

and p H = vH −

Q − KqL α(qH − qL )

Its optimal profit is

πKQ

1 ≡ qH − q L

  Q2 + E(q 2 )K 2 − 2KQE(q) (vH − vL )Q + (vL qH − vH qL )K − α(1 − α)(qH − qL )

Primal and dual feasibility of this local optimum require that max (Q/qH , g(Q)) ≤ K ≤ min (Q/qL , f (Q)) Case 4 Clearly, the global unconstrained optimum (λ1 = 0 and λ2 = 0) is attainable whenever K ≥ E(v)/2 and Q ≥ E(vq)/2. It consists of the firm choosing prices vL /2 and vH /2, and its value is πU = α

2 vH v2 + (1 − α) L 4 4

I characterize the monopoly’s optimal pricing strategy by comparing the local maxima obtained above for all possible range of parameters. Figure 1 depicts the firm’s optimal division of the K-Q space.5 The monopoly’s optimal behavior is different in each of these regions. The firm chooses optimal prices in such a way that only capacity K binds in region K (Case 1), only capacity Q binds in region Q (Case 2), both K and Q bind in region KQ (Case 3). None of the constraints bind in region U which corresponds to the unconstrained optimum (Case 4). Only capacity K binds and the monopoly excludes low-types in region EL. Conversely, only capacity Q binds and the monopoly excludes high-types in region EH. Notice that each of the 4 lines bordering the core region KQ have a different slope, hence the region KQ is not symmetric. Proposition 1 provides a complete characterization of the firm’s optimal choice for any combination of its capacity levels.   Figure 1 depicts the case of (1 − α)qL vL − vH qqHL < αqH (vH − vL ). When this ordering is reversed the figure changes accordingly. However, the coordinates of all lines and critical points remain the same. 5

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Figure 1: Division of the capacity-space based on the monopoly’s optimal strategy

K

Q

EH

U

E(v) 2

f (Q)

K 1−α 2 (vL

g(Q)

− vH qqHL ) L α vH −v 2

KQ Q/qH Q/qL

1−α 2 qL (vL

EL − vH qqHL )

L αqH vH −v 2

E(vq) 2

Q

Proposition 1. The optimal prices of the monopoly are L L • pL = vL + α vH −v − K and pH = vH − (1 − α) vH −v − K in region K where 2 2 K binds, Q is slack, some consumers of both groups are served;   qL qH α • pL = E(q (v + v )q + (1 − α)v q − Q and pH = pL qqHL in reL qL H H L L 2) 2 gion Q where Q binds, K is slack, some consumers of both groups are served;

KqH −Q Q−KqL and pH = vH − α(q in region KQ where both K • pL = vL − (1−α)(q H −qL ) H −qL ) and Q bind, some consumers of both groups are served;

• pH = vH − • p L = vL − excluded;

K α

in region EL where K binds, Q is slack, low-types are excluded;

Q qL (1−α)

in region EH where Q binds, K is slack, high-types are

• pL = vL /2 and pH = vH /2 in region U where both K and Q are slack, some consumers of both groups are served.

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Assumption 1 guarantees the existence of all 6 regions enumerated in Proposition 1. The main intuition driving the results is the two consumer groups’ varying relative attractiveness for the firm. For any given Q, an increase in K makes the constraint on total production, Q, tighter. This in turn makes low-types, that consume less of the tighter capacity, relatively more valuable for the firm, so in optimum the firm adjusts its prices to attract more of them (except for extreme values of K). Figure 2: Small Q rH Prices

1 vH pH

vH −

Q αqH

vL pL vL −

Q (1−α)qL

0 Q qH

Q qL

K

(a) Prices

3.2

Q qH

Q qL

K

(b) Share of high-types

Comparative statics

In this section I investigate the comparative statics properties of the monopoly’s optimal behavior. The variables of interest are optimal prices charged for the two consumer groups and the share of high-types among all consumers served. I present all the results as a function of the constraint on the number of people served, K, however, analogous statements can be made for the other constraint, Q, as the model is almost symmetric in K and Q. For a detailed analysis of the firm’s pricing behavior, it will be useful to divide the Q-K plane into 4 regions by 3 vertical lines going through the 3 values that appear on the Q axis of Figure 1. I call the 4 resulting cases “small Q”, “medium Q”, “large Q”, and “very large Q”. 3.2.1

Small Q

The first region of interest is delimited by    1−α qL 0 < Q ≤ min qL vL − vH , 2 qH 16

vH − vL αqH 2



Figure 3: Medium Q, Case A rH Prices

1

vH pH

vH −

Q αqH

pL

vL

Q qH

f (Q)

Q qL

K

0

(a) Prices

Q qH

f (Q)

Q qL

K

(b) Share of high-types

For very low levels of K the firm excludes low-types and sells up to K exclusively to high-type consumers, Q is slack here. As K grows larger, the monopoly starts serving some low-types as well in such a way that both capacity constraints bind. This means that as K grows the firm serves more and more low-types and less and less high-types. Finally, as K becomes even larger high-types get excluded and the firm only serves low-types. The intuition is the following. As K grows, Q, the constraint on the production gets relatively stricter. This means that low-type consumers who use up less of the production constraint become more and more valuable for the firm. This is indeed reflected in the share of high-type consumers served decreasing from 1 all the way to 0 as depicted in the right panel of Figure 2. It is interesting to note that while pL is decreasing in K, pH first decreases then increases (see panel a of Figure 2). The decrease occurs in region EL where only high-types are served: the monopoly decreases its prices so that the demand of high-types equals the capacity level. The increase is again a consequence of low-types becoming more valuable for the firm: In order to accommodate more low-types, the firm must serve less high-types so it raises its price for them.

3.2.2

Medium Q

The monopoly’s pricing strategy changes somewhat when Q is of intermediate size, i.e.,

17

Figure 4: Medium Q, Case B rH Prices

1 vH vL +vH 2

pH

vH /2 vL pL

0 L α vH −v 2

g(Q)

K

Q/qL

(a) Prices

 min

L α vH −v 2

g(Q) Q/qL

K

(b) Share of high-types

       1−α vH − vL 1−α vH − vL qL qL qL vL − vH , αqH < Q ≤ max qL vL − vH , αqH 2 qH 2 2 qH 2 

Case A: (1 − α)qL vL −

vH qqHL



< αqH (vH − vL ) :

Case A represents the parameter setting depicted in Figure 1. As before, for very low levels of K the firm excludes low-types. For larger levels of K the firm starts serving low-types and serves less and less high-types. However, when K hits the threshold value of f (Q), the prices become independent of the size of K. For these capacity levels the firm serves some consumers of both groups in such a way that Q binds and K is slack. Intuitively, as K grows serving low-types becomes relatively more profitable so there is a region where the mass of high-types served decreases. The profit is always the sum of the profit the firm makes on serving low-types and high-types. It is easy to see that both of those partial profit curves are concave. The profit the monopoly earns on high-types is decreasing at an increasing rate while the profit made on low-types is increasing at a decreasing rate. This means that there is a point (f (Q)) where the marginal revenue of high-types equals the marginal loss on low-types. At this point the monopoly prefers to switch to the Q regime where the share of high-types does not decrease more in K. As f (K) < Q/qL , the share of high-types does not decrease all the way to 0 (see Figure 3).

18

Figure 5: Large Q rH 1

Prices vH vL +vH 2

pH

vL pL

0 L α vH −v 2

g(Q)

f (Q)

K

(a) Prices

L α vH −v 2

g(Q) f (Q)

K

(b) Share of high-types

  Case B: αqH (vH − vL ) ≤ (1 − α)qL vL − vH qqHL : As before, for very low levels of K the firm excludes low-types. However, as K grows the capacity levels enter the K region which means that K still remains binding while both types start getting served and both prices keep decreasing in K. Next, K reaches the KQ region where the monopoly starts raising pH and both constraints bind. The increase in K continues until none of the high-types are willing to buy, only low-types are served and Q binds. The mass of high-types firstly increases with K, then the mass still increases but their share starts decreasing hyperbolically. When both constraints bind even the absolute mass of high-types starts decreasing so their share drops even further, according to a steeper hyperbola until it reaches zero, see Figure 4. 3.2.3

Large Q

The next region of interest is    1−α qL max qL vL − vH , 2 qH

vH − vL αqH 2


E(v) 2

The evolution of optimal prices and consumer shares is very similar to the one for medium Q, case B, as one can see from Figure 5. Indeed, as K grows the optimal prices are chosen from regions EL, K, KQ, respectively. However, as Q is larger here, the monopoly does not exclude high-types even for very large values of K, the last

19

Figure 6: Very large Q Prices rH

vH 1 vL +vH 2

pH

vH /2 vL pL

vL /2 L α vH −v 2

E(v)/2

K

αvH /E(v) 0

(a) Prices

L α vH −v 2

E(v)/2

K

(b) Share of high-types

region the prices are chosen from is Q. Accordingly, the share of high-types varies as described in Case B up to K = f (Q) where it becomes independent of K and levels off at a strictly positive value. 3.2.4

Very large Q

When Q > E(vq)/2, the capacity constraint on the amount of production will never be binding.6 For very low levels of K the firm excludes low-types and serves only high-type consumers. As K grows larger the monopoly starts to serve some low-types as well, although the constraint on the number of consumers still binds. Finally, as K becomes very large neither of the constraints will bind and the monopoly can achieve the unconstrained optimum. Clearly, both prices pH and pL are weakly decreasing in K in this region (Figure 6a). The share of high-types served decreases from 1 to αvH /E(v) which corresponds exactly to the proportion of high-types in the whole market (Figure 6b).

4

Welfare

In this section, I investigate welfare properties of the monopoly’s optimal pricing behavior in the presence of dual capacity constraints. I first analyze aggregate consumer surplus as a function of the capacity constraints given that the monopoly 6

One can see the case of very large Q as an alternative benchmark. As Q never binds, this region describes the monopoly’s optimal strategy when there is only one capacity constraint and two consumer groups.

20

chooses its profit-maximizing prices described in Proposition 1. Next, I calculate total welfare as the sum of consumer surplus and the monopoly’s profit, although the most interesting insights come from the study of consumer surplus. Figure 7: Non-monotonicity of consumer surplus

K

Q

EH

U

E(v) 2

K 1−α 2 (vL

− vH qqHL )

KQ

L α vH −v 2

EL

Q/qH 1−α 2 qL (vL

− vH qqHL )

L αqH vH −v 2

E(vq) 2

Q

When both consumer groups are served, the general form of total consumer surplus writes as α 1−α (vH − pH )2 + (vL − pL )2 . 2 2 In the EL region, where the monopoly only serves high-types, consumer surplus equals α2 (vH − pH )2 . In the EH region, where the firm serves exclusively the low-types, consumer surplus equals 1−α (vL − pL )2 . 2 CS =

Thus consumer surplus is weakly decreasing in both prices pL and pH . As shown in the previous section, with the exception of the KQ region where both constraints bind, prices are decreasing in the size of capacities. Hence consumer surplus is 21

increasing in capacity levels for any capacity-pairs outside of the KQ region. However, the problem is more complicated in the KQ region where both capacities bind. In this region pH is increasing while pL is decreasing in K. The following proposition sheds light on the effect of this trade-off. Proposition 2. Aggregate consumer surplus is non-monotonic in the size of the capacity constraints in the parameter region where both capacities are binding. In particular, there always exists a region of capacity pairs inside KQ where consumer surplus is decreasing in K and increasing in Q. Moreover, there exists a second region inside KQ where consumer surplus is decreasing in Q and increasing in K. The two regions are always disjoint. The proof of Proposition 2 shows that consumer surplus is decreasing in K in E(q) the KQ region if and only if K ≤ Q E(q 2 ) (light grey area in Figure 7). The intuition for this result is the clearest when Q is relatively low and one can consider the transition between the regions EL and KQ. When K is increasing from a value lower than Q/qH the monopoly first excludes all low-types and serves only high-types. As K reaches Q/qH it starts to be profitable to serve some low-type consumers as well such that both constraints bind. To accommodate low-types, the firm starts serving less high-types thus it increases pH while it lowers pL . Consumer surplus is affected by 3 factors. Firstly, as K is binding in both regions, the total mass of consumers served goes up which ceteris paribus increases consumer surplus. Secondly, the decrease in pL has the same effect also, however, the increase of pH goes in the opposite direction. To see that this last effect dominates the first two, one should consider the mass of consumers affected. Indeed, when K is relatively small, the firm serves a lot more high-types than lowtypes, so the loss suffered by the high-types dominates the gain of the few low-types. Similarly, consumer surplus is decreasing in Q in the KQ region if and only if K ≥ Q/E(q) (dark grey area in Figure 7) and the arguments are analogous to the ones described above. The next proposition provides comparative statics results for total welfare, which in this context simply equals to the sum of the monopoly’s profit and consumer surplus.

Proposition 3. Total welfare, i.e., the sum of consumer surplus and the monopoly’s profit, is increasing in both capacity levels for every parameter value. I show in the Appendix that in both areas where consumer surplus is decreasing, profit increases faster than consumer surplus decreases. This property is obviously 22

true for other capacity pairs where both consumer surplus and profit are increasing in capacities.

5

Endogenous capacity choice

In this section I investigate the monopoly’s optimal choice of capacity levels in the long run. Although in the short run it is reasonable to assume that the monopoly chooses its prices facing fixed capacity levels, in the long run firms can extend or shrink both of their capacities. Let cK (K) denote the cost of building capacity K and let cQ (Q) denote the cost of building Q. I assume that the costs are separable. In the hospital example, although there is a fixed cost of constructing the hospital building, the additional costs of adding beds and equipping the operating rooms are separable. Assume that these costs are strictly positive whenever the capacity levels are strictly positive. The monopoly maximizes its profit which is now a function of its two capacities as well as its prices. The optimal choice of prices for any capacity-pair is the one described in Proposition 1. The following Lemma holds under these very general conditions. Lemma 1. If capacity choice is endogenous and the cost of building capacities is strictly positive then the monopoly chooses its prices and capacity levels in such a way that both constraints bind, i.e., the optimal capacities are always chosen from the KQ region. An alternative interpretation of Lemma 1 is that the monopoly never chooses capacities in such a way that only one of the capacities be binding. Intuitively, in a world of deterministic demand it is never profitable for a monopoly to build unused capacity. Notice that choosing capacities from the KQ region does not necessarily mean that the monopoly serves both types of consumers. The firm may choose its capacities at the limit of the region in a way to exclude all consumers of one or the other type. In order to obtain closed-form results and hence a clear intuition, I will focus on the case of linear costs in the remainder of the section, i.e., let cK (K) = cK

and cQ (Q) = dQ.

23

The following proposition provides a complete characterization of the monopoly’s optimal capacity choice given the linear cost functions. Proposition 4. If cost functions are linear, i.e., cK (K) = cK then the optimal capacity levels are given by

and cQ (Q) = dQ

1. K = 21 (E(v) − c − dE(q)) and Q = 12 (E(vq) − cE(q) − dE(q 2 )) if vH > c + dqH and vL > c + dqL 2. K = α2 (vH − c − dqH ) and Q = α2 qH (vH − c − dqH ) if vH > c + dqH and vL ≤ c + dqL (vL − c − dqL ) and Q = 1−α qL (vL − c − dqL ) 3. K = 1−α 2 2 if vH ≤ c + dqH and vL > c + dqL 4. K = Q = 0 if vH ≤ c + dqH

and vL ≤ c + dqL

The resulting optimal prices are pi = (vi + c + dqi )/2, i-types are not excluded.

i ∈ {L, H} whenever

Notice that c + dqH (c + dqL ) corresponds to the marginal cost of building capacity to serve an additional high-type (low-type) consumer. Some high-type consumers are served if and only if the marginal cost of capacity necessary to serve a high-type is lower than the WTP of the most valuable high-type consumer, i.e., vH > c + dqH . An analogous result holds for low-types. Hence the four cases of Proposition 4: depending on the relative sizes of the two marginal costs with respect to the WTP of the most valuable consumers, the firm either serves both types or excludes one or both groups of consumers. The first case in Proposition 4 corresponds to the situation when some consumers of both types are served, i.e., capacities are chosen from the inside of the KQ region. The second case describes a situation where the monopoly prefers excluding low-type consumers. This arises when the capacity constraint on production, Q is relatively cheap to build, which makes K relatively stricter which in turn increases the attractively of high-types. Indeed, the two conditions imply d < (vH − vL )/(qH − qL ). The optimal capacities satisfy K = Q/qH which means that the monopoly chooses its capacities from exactly one side of the KQ quadrilateral. Notice also that the resulting price for high-types, pH = (vH + c + dqH )/2, corresponds exactly to the optimal price of a monopoly facing only high-type consumers and whose production costs are equal to c + dqH .

24

The third case is analogous to the second one: the firm prefers excluding hightypes and chooses its capacity from the K = Q/qL side of the KQ quadrilateral. The fourth case completes the discussion: if both capacities are very expensive to build with respect to the WTP of all consumers then the monopoly prefers to exit the market. As one should expect, capacity levels are decreasing in costs, moreover, transitions between the different cases are smooth in the sense that K and Q are continuous in both c and d. In the limit as both costs go to 0, the monopoly builds enough capacities to achieve its unconstrained optimum, i.e., K → E(v)/2 and Q → E(vq)/2. Both prices pL and pH are an increasing function of both cost parameters c and d. Therefore, the consumer surplus is always decreasing in both c and d when capacities are chosen endogenously. Hence, in the long run both prices and consumer surplus are monotonic in capacity costs.

6

Incentive compatibility

In this section I relax the assumption that the monopoly is able to observe the quantity demanded by each consumer. Given the self-selection of consumers, the firm faces new incentive compatibility constraints. I investigate two different extensions of the baseline model. In the first scenario low-types are allowed to buy the large bundle and freely dispose of the unused part. In the second scenario, I check the robustness of the baseline model by assuming that qH is a multiple of qL , and that high-types are allowed to buy several small bundles .

6.1

Incentive compatibility for low-types

In this section I consider an extension of the model where low-types are allowed to buy quantity qH for pH and throw away the quantity qH − qL they do not consume. One can think of this scenario as the case with free disposal. This assumption is realistic if the monopoly cannot tell consumers apart according to their demand. How does consumers’ possibility of free disposal alter the monopoly’s incentives? Firstly notice that free disposal only alters consumers’ (low-types’) incentives in case pH < pL ≤ vL , i.e., whenever the larger quantity is cheaper. Problem (P-EH) where only low-types are served will thus remain unaffected by free disposal.

25

The maximization problem of serving both consumer groups becomes more complicated in the presence of free disposal. The monopoly must decide whether it chooses its prices in a way that makes both types buy the quantity intended for them, or alternatively, it may choose a price such that all consumers buy the larger quantity, qH . In the former case, the maximization problem is very similar to (P-LH) described previously, with one additional constraint:

(P-ICL)

max

pL ,pH

π = α(vH − pH )pH + (1 − α)(vL − pL )pL

s.t.

α(vH − pH ) + (1 − α)(vL − pL ) ≤ K α(vH − pH )qH + (1 − α)(vL − pL )qL ≤ Q 0 ≤ pL < vL

, pL ≤ pH < vH

The monopoly must choose a lower price for the smaller quantity if it wants the two consumer groups separated, hence the new, stricter lower bound on for the high-price: pL ≤ pH . However, we know from the previous section that the solution of (P-LH) for any parameter region satisfies this stricter condition. This means that the solution of (P-ICL) will exactly coincide with the solution of (P-LH) described by Proposition 1. In addition, the monopoly might now choose a price pH < pL ≤ vL that makes all consumers buy qH for pH . In this case the mass of buyers is α(vH − pH ) + (1 − α)(vL − pH ) = E(v) − pH and total demand is equal to (E(v) − pH )qH . Hence the maximization problem writes as

(P-ICL2)

max π = (E(v) − pH )pH pH

s.t.

E(v) − pH ≤ min(K, Q/qH ) pH < vL

Notice that the two constraints imply that the following two inequalities must be satisfied for (P-ICL2) to have a feasible solution: K > α(vH − vL ) and Q > αqH (vH − vL ) 26

Intuitively, the capacity constraints must be relatively large if the monopoly is able to serve some consumers of both types for a relatively low price. Finally, if the monopoly wants to exclude low-types from buying its products, it must satisfy the stricter condition of vL < pH :

(P-ICLEL)

max π = α(vH − pH )pH pH

s.t.

α(vH − pH ) ≤ min(K, Q/qH ) vL < pH However, the optimal solution of (P-ICLEL) is obviously dominated by the solution of (P-EL) as it is a constrained version of it. Proposition 5. The monopoly’s optimal pricing strategy is not affected by the possibility of free disposal. Proposition 5 states that the model of dual capacities is robust to the introduction of the free disposal assumption. The new strategies of the monopoly consist of pooling both types at pH and selling them the large quantity qH . The proof in the Appendix shows that this pooling is always dominated by some strategy that was already available in the baseline model.

6.2

Incentive compatibility for high-types

In this section I consider a model in which the individual demanded of high-types is a multiple of the individual demand of low-types, i.e., qH = kqL , where k is an integer. The monopoly cannot observe consumers’ individual demand. High-types therefore must make a choice between buying one bundle of qH or k bundles of qL . The monopoly must decide whether to choose prices such that high-types prefer buying the large bundle (qH ) or to choose them in a way that everyone buys small bundles (qL ). Given that the monopoly is unable to tell apart consumers, it must choose a low enough price for qH if it wants high-types to buy it instead of several small bundles. In particular, the unit price of the large bundle cannot exceed the unit price of the small bundle: pH ≤ kpL ⇐⇒ pH /qH ≤ pL /qL . Notice that the optimal solutions of (P-LH), as described in Proposition 1 satisfy this condition for any capacity-pair. Hence, similarly to the free disposal case, the additional incentive compatibility constraint does not alter the firm’s optimal 27

pricing behavior if it wants the two consumer groups to buy different bundles. The other option of the firm is to choose a relatively low price for the small quantity, resulting in all consumers buying that variety. The total price a high-type buyer must pay to satisfy its demand qH is kpL . The maximization problem when serving some consumers of both types then writes as

(P-ICH)

pL pL )qH + (1 − α)(vL − pL )pL s.t. qL qL pL α(vH − qH ) + (1 − α)(vL − pL ) ≤ K qL pL α(vH − qH )qH + (1 − α)(vL − pL )qL ≤ Q qL 0 ≤ kpL < vH

max π = α(vH − qH pL

The last inequality ensures a low enough price so that some high-types consume the product. Notice that this maximization problem coincides with (P-LH) with the additional constraint of pH = kpL = qH pqLL . As shown above, the monopoly’s optimal choice with perfect divisibility coincides with the solution of (P-LH). (P-ICH) being a more restricted problem, it can never be more profitable for the firm to make all consumers buy the small bundle than to separate the two consumer groups. Finally, the monopoly can choose to exclude one group of consumers. The firm faces exactly the same problem as (P-EL) if it serves only qH bundles to hightypes. Serving only qL bundles does not necessarily exclude the high-types, if the monopoly wants to serve only low-types, it must choose a relatively high unit price: vH < pqLL < vqLL . However, the solution of this sub-problem either coincides with the qH solution of (P-EH), or it is dominated by serving both consumer groups, hence the following proposition: Proposition 6. The monopoly’s optimal pricing strategy is not affected by high-type consumers’ ability to buy several small bundles despite the monopoly’s inability to observe consumers’ demand. Proposition 6 concludes the analysis of incentive compatibility. Jointly with Proposition 5, it suggests that the results of the model are robust to the relaxation of the observable individual demand assumption.

28

7

General distribution of consumers

In the baseline model presented in Section 2, consumers’ WTP is distributed uniformly within each group, which implies linear demand that in turn leads to the clear-cut results presented in Proposition 1. In this section I generalize the baseline model by assuming a general distribution function of consumers’ WTP. I identify sufficient conditions on the distribution functions of the two consumer groups that guarantee the existence of the regions discovered in the baseline model, illustrated by Figure 1. Moreover, I show that under these conditions the main results carry over to the general distribution case. Let the WTP of consumers of type i ∈ {L, H} be distributed according to the twice continuously differentiable distribution Fi with support [0, θi ]. Let fi denote the first derivative of Fi . For any p ≥ 0 let Di (p) = 1−Fi (p) be the demand function that measures the proportion of i-type consumers willing to buy at price p. Let α and 1−α denote the total mass of high-types and low-types, respectively. Individual demand of consumers is the same as in the baseline model: qH > qL . As before, the monopoly is constrained by K, the total mass of consumers it can serve on the one hand, and by Q, its maximal production on the other hand. The maximization problem of the monopoly writes as

(P-GEN)

max

pL ,pH

π = αpH DH (pH ) + (1 − α)pL DL (pL )

s.t.

αDH (pH ) + (1 − α)DL (pL ) ≤ K

(λ1 )

αqH DH (pH ) + (1 − α)qL DL (pL ) ≤ Q

(λ2 )

pL ≥ 0 , pH ≥ 0

where λ1 is the multiplier of the constraint on the mass of consumers served and λ2 is the multiplier of the constraint on total production. The only restrictions on the demand functions so far are that they be decreasing, Di (0) = 1 and Di (θi ) = 0. Notice that Di (pi ) = 0 for any pi ≥ θi , so choosing a large enough pi excludes consumer group i. Assume that the profits derived from serving the two groups, i.e., pi Di (pi ), is concave. The monotone hazard rate condition, which in this context translates to the function φi (p) ≡

fi (p) D0 (p) =− i 1 − Fi (p) Di (p)

being non-decreasing, is a sufficient condition for the concavity of pi Di (pi ). The Karush-Kuhn-Tucker conditions of the maximization problem write as 29

0 (pL −λ1 −λ2 qL )DL0 (pL )+DL (pL ) = 0 and (pH −λ1 −λ2 qH )DH (pH )+DH (pH ) = 0.

These conditions are trivially satisfied for a consumer group which is excluded from the market. Otherwise, when prices are chosen such that some consumers of both types are served (i.e., pL < θL and pH < θH ) the first order conditions can be rewritten using the distribution functions:

λ1 + λ2 q L = pL −

1 − FL (pL ) fL (pL )

and λ1 + λ2 qH = pH −

1 − FH (pH ) . fH (pH )

i (p) Notice that the term p − 1−F which appears in both equations is the virtual fi (p) valuation function that is widely used in the mechanism design literature.7

The first order conditions imply that in optimum the virtual valuation of the two consumer groups must coincide whenever the capacity on the number of people served (K) binds and the other constraint is slack (λ1 > 0, λ2 = 0). Conversely, if Q binds and K is slack (λ1 = 0, λ2 > 0) then the per-unit virtual valuation of the two groups must be equal. The monopoly can charge its unconstrained optimal prices (where both virtual valuations equal zero) if none of the constraints bind. Replacing the optimal prices into the two capacity constraints, one gets a unique threshold level for both capacity levels, K and Q that delimit region U’. Next, I identify sufficient conditions for the existence of a core region KQ’ where both constraints bind. Both constraints binding immediately imply that the optimal prices must satisfy DL (pL ) =

KqH − Q (1 − α)(qH − qL )

and DH (pH ) =

Q − KqL . α(qH − qL )

This means that 2 of the 4 curves delimiting the core region KQ’ remain the same as in the baseline model: in order to serve both consumer groups, K ≥ Q/qH and K ≤ Q/qL must be satisfied. The two other frontiers of the KQ’ region are given by capacity-pairs within the region that also satisfy the equations λ1 = 0 and λ2 = 0. Let K = g(Q) denote the curve above which capacity K is slack (λ1 = 0), and let K = h(Q) be the curve that specifies whether Q is binding or slack (λ2 = 0). Finding an explicit formula for these curves would necessitate knowing the value of Di0 (pi ) for the optimal prices. Although demand levels at the optimal prices are 7

The connection between auction theory and the monopoly’s problem of third-degree price discrimination was first revealed by Bulow and Roberts (1989).

30

simple, there is no direct formula to express the derivatives of the demands at the optimal prices. However, using the implicit function theorem, one can prove the following Lemma. Lemma 2. (i) Assume that both distribution functions satisfy Myerson’s regularity condition, i (p) i.e., the virtual valuation functions p − 1−F , i ∈ {L, H} are strictly infi (p) 0 0 creasing in p. Then g (Q) > 0 and h (Q) > 0. (ii) g(Q) < h(Q) for any 0 ≤ Q < Q and g(Q) = h(Q) = K. e > 0 such that g(Q) ≤ Q/qH for ∀ 0 ≤ Q < Q. e (iii) ∃ Q e > 0 such that h(Q) ≥ Q/qL for ∀ 0 ≤ h(Q) < K. e (iv) ∃ K (i) identifies regularity of the distribution functions as sufficient conditions for curves g and h to be increasing. Notice that the monotone hazard rate condition implies Myerson’s regularity condition. (ii) states that g is always below h when at least one of the capacities are binding, and they cross exactly at point (Q, K). (iii) and (iv) state that g(Q) < Q/qH for small values of Q and conversely, h(Q) > Q/qL for small values of K. The combination of these results imply the existence of a KQ’ region delimited by two increasing curves in addition to the two straight lines. Next, I show that the existence of region EL’ where the monopoly excludes low-types is guaranteed if the distribution of high-types hazard rate dominates the distribution of low-types, i.e., fH (p) fL (p) < 1 − FH (p) 1 − FL (p)

for all p.

Notice that the hazard rate dominance relation rewrites in terms of demand as εH (p) < εL (p) for all p, where εi (p) is the elasticity of demand: pDi0 (p) . εi (p) = − Di (p) Therefore, hazard rate dominance is equivalent to the low-types having a larger demand elasticity than high-types at each price. As hazard rate dominance implies first order stochastic dominance, it also implies θL < θH . The proof consists of showing that firstly, prices are decreasing and demands are increasing in K in the K’ region, and secondly, there is a positive threshold level of K below which optimal prices induce zero demand for the low-types and strictly positive demand for the high-types. Intuitively, the curve separating regions K and EL is a horizontal line because in both regions only constraint K binds and the prices are profits are 31

independent of Q. Conversely, I show that a sufficient condition for the existence of an EH’ region where the monopoly serves exclusively the low-type consumers is

qL

fL (p) fH (p) < qH 1 − FL (p) 1 − FH (p)

⇐⇒

qL εL (p) < qH εH (p) for all p.

The sufficient condition requires that for every price, the demand elasticity weighted by the individual demand be higher for the high-types than for the lowtypes. This ensures the existence of a threshold level of Q at which the optimal prices in the Q’ region induce a zero demand for the high-types while demand of low-types remain positive. The curve separating regions Q’ and EH’ is a vertical line because prices and profits in both regions only depend on Q and are independent of K. Consumer surplus in the context of general distributions can be written as Z

θH

Z

θL

(w − pH )fH (w)dw + (1 − α)

CS = α pH

(w − pL )fL (w)dw. pL

Obviously, the consumer surplus is a decreasing function of both prices. In most regions both prices are decreasing in both capacity levels, thus aggregate consumer surplus is increases in both K and Q. However, in the KQ’ region pH is increasing in K and decreasing in Q and conversely, pL is increasing in Q and decreasing in K. Therefore, consumer surplus of the high-types is decreasing while consumer surplus of low-types is increasing in K. The following proposition states that there always exists a region of capacity-pairs inside of KQ’ where the first effect is dominant. Proposition 7. Assume that both distribution functions satisfy the monotone hazard rate condition. Then there exist two disjoint regions inside of KQ’ such that the consumer surplus decreases in K in one of the regions, and it decreases in Q in the other region. The monotone hazard rate condition implies both the concavity of profits and increasing virtual valuation functions. In the Appendix, I show that ∂CS DL (pL ) DH (pH ) < 0 ⇐⇒ 0 < 0 ∂Q DL (pL ) DH (pH )

and

∂CS D0 (pL ) D0 (pH ) < 0 ⇐⇒ qL L < qH H . ∂K DL (pL ) DH (pH )

It follows from Lemma 2 that part of the Q/qL and Q/qH lines border the KQ’ region. From the demand functions, it is obvious that the first condition is satisfied for K = Q/qL and the second for K = Q/qH . Existence of the two regions can be 32

proved by continuity arguments. In order to gain intuition for the above formulas, one can rewrite them in terms of elasticities: ∂CS εL (pL ) pL < 0 ⇐⇒ < ∂Q εH (pH ) pH

and

∂CS εL (pL ) pL /qL < 0 ⇐⇒ > . ∂K εH (pH ) pH /qH

Consumer surplus in decreasing in Q whenever at the optimal prices the demand elasticity to price ratio is larger for high-types than for low-types. Furthermore, consumer surplus is decreasing in K when at the optimal prices the demand elasticity to unit price ratio is higher for the low-types than for the high-types. This formulation also shows clearly that there is no capacity-pair that satisfies both conditions, i.e., there is no situation in which an increase in both capacities decreases consumer surplus.

8

Conclusion

Several capacity constraints co-exist in various real-world industries. The present paper provides a formal economic analysis of the effects of dual capacity constraints on optimal firm behavior. It reveals a rich structure of optimal monopoly pricing which in the short run is qualitatively different from the predictions of models of firms bound by a single capacity. In particular, prices charged for some consumers increase for some capacity pairs as one of the capacities is enlarged. Moreover, aggregate consumer surplus is decreased by an increase of one capacity level for some capacity pairs. These results are robust to observability of individual demand and also for a fairly general class of distribution of consumers. In the long run, when capacity building is endogenous, prices and consumer surplus are monotonic in capacity costs. Future research can extend the results in several important aspects. One could verify the model’s robustness by approximating the capacity constraints with convex and continuous cost functions. Moreover, the model could accommodate more than two consumer groups. Finally, by extending the model to the case of a duopoly, it would become directly comparable with the various models of Bertrand-Edgeworth competition with one capacity constraint.

33

Appendix Proof of Proposition 1 Consider the following maximization problem that encompasses (P-LH), (P-EL) and (P-EH):

(P-GEN)

max

pL ,pH

π = α(vH − pH )pH + (1 − α)(vL − pL )pL

s.t.

α(vH − pH ) + (1 − α)(vL − pL ) ≤ K

(λ1 )

α(vH − pH )qH + (1 − α)(vL − pL )qL ≤ Q

(λ2 )

pL ≤ v L

(λ3 )

pH ≤ v H

(λ4 )

pL ≥ 0 , pH ≥ 0 The non-negativity constraints are omitted and verified ex-post. Multiplying the third and fourth constraint by 1 − α and α, respectively, the objective function for deriving the Karush-Kuhn-Tucker conditions writes as

L(pL , pH , λ1 , λ2 , λ3 , λ4 ) = α(vH − pH )pH + (1 − α)(vL − pL )pL − λ1 [α(vH − pH ) + (1 − α)(vL − pL ) − K] − λ2 [α(vH − pH )qH + (1 − α)(vL − pL )qL − Q] − λ3 (1 − α)(vL − pL ) − λ4 α(vH − pH ) Hence the two first order conditions are 2pL = vL + λ1 + λ3 + λ2 qL

and 2pH = vH + λ1 + λ4 + λ2 qH .

As λ3 > 0 implies pL = vL , this case corresponds to excluding the low-types. Indeed, (P-GEN) is reduced to (P-EL) whose solution is described in the main text. Similarly, λ4 > 0 implies pH = vH i.e., the exclusion of high-types which corresponds to the maximization problem (P-EH), also solved in the main text. Obviously, λ3 > 0 and λ4 > 0 leads to zero profit hence it is never optimal. Finally, λ3 = λ4 = 0 corresponds to the case of serving both consumer groups, described in (P-LH). In the following, I prove the formulas for optimal prices and the borders of the optimal regions described in Case 1 - Case 3. Case 4 is described in the main body of the paper.

34

Case 1 : λ1 > 0 and λ2 = 0 From the FOCs: λ1 = 2pL − vL = 2pH = vH and λ1 > 0 implies that K binds: α(vH − pH ) + (1 − α)(vL − pL ) = K. The optimal prices can be calculated from these 2 equations: vH − vL vH − vL − K and pH = vH − (1 − α) −K 2 2 The capacity constraint on production rewrites as pL = v L + α

vH − vL vH − vL )qH + (1 − α)(K − α )qL ≤ Q, 2 2 which is equivalent to K ≤ g(Q) by definition. λ1 > 0 implies K < E(v)/2 and finally, pL ≤ vL implies α2 (vH − vL ) ≤ K. The non-negativity constraints and pH ≤ vH are always satisfied in this optimum. α(K + (1 − α)

Case 2 : λ1 = 0 and λ2 > 0 L From the FOCs: λ2 = 2pLq−v = 2pHqH−vH and λ2 > 0 implies that Q binds: L α(vH − pH )qH + (1 − α)(vL − pL )qL = Q. The optimal prices can be calculated from these 2 equations. The capacity constraint K must be slack, replacing the optimal  prices into the constraint leads to f (Q) ≤ K. Moreover, pH ≤ vH implies 1−α qL vL − vH qqHL ≤ Q and finally, λ2 > 0 implies Q < E(vq)/2. 2

Case 3 : λ1 > 0 and λ2 > 0 Both K and Q bind, hence the optimal values of the prices follow directly from the two equations. The four borders of the KQ regions can be calculated as follows. The values of the two multipliers can be calculated from the FOCs by replacing the optimal prices: λ2 = 2

vH − vL Q − KqH KqL − Q + − 2 qH − qL α(qH − qL ) (1 − α)(qH − qL )2

and

  Q − KqL vH − vL Q − KqH KqL − Q λ1 = 2vH − 2 − qH 2 + − . α(qH − qL ) qH − qL α(qH − qL )2 (1 − α)(qH − qL )2 It follows that λ1 > 0 is equivalent to K ≥ g(Q) and λ2 > 0 is equivalent to K ≤ f (Q). Furthermore, pL ≤ vL implies K ≤ Q/qL and pH ≤ vH implies K ≥ Q/qH .

35

Excluding one consumer group In the parameter regions delimited by one of the Cases 1 - 4, the optimal prices are such that some consumers of both groups are served. There are two remaining regions where one group of consumers will be excluded. If  K ≤ min Q/qH , α2 (vH − vL ) then K must bind, so one must compare the profit 2 K2 of vH K − Kα when excluding low-types with the profit of vL K − 1−α obtained by α excluding the high-types. K ≤ 2 (vH −vL ) implies that the first profit, i.e., excluding the low-types is always more profitable for small values of K. An analogous argument shows why is more profitable than excluding low-types  excluding high-types  qL 1−α when Q ≤ min KqL , 2 qL vL − vH qH .  Proof of Proposition 2 Replacing the optimal prices pL and pH of region KQ into the general formula, the consumer surplus equals 2  2 Q − KqL KqH − Q 1−α + = α(qH − qL ) 2 (1 − α)(qH − qL )  1 2 2 2 = Q − 2E(q)KQ + E(q )K . 2α(1 − α)(qH − qL )2

α CS = 2



Therefore the first derivative of the consumer surplus with respect to K and Q are  1 ∂CS 2 = −E(q)Q + E(q )K ∂K α(1 − α)(qH − qL )2 ∂CS 1 = (−E(q)K + Q) . ∂Q α(1 − α)(qH − qL )2

and

Thus consumer surplus is decreasing in K whenever K ≤ E(q)/E(q 2 )Q and it is decreasing in Q if K ≥ Q/E(q). The two regions delimited by these lines are disjoint since they both cross the origin and the slope of K = E(q)/E(q 2 )Q is smaller than the slope of K = Q/E(q) as (E(q))2 < E(q 2 ).  Proof of Proposition 3 Total welfare is given by the sum of consumer surplus and profit:

36

1 ((vH − vL )Q + (vL qH − vH qL )K) − (qH − qL )  1 − Q2 − 2E(q)KQ + E(q 2 )K 2 . 2 2α(1 − α)(qH − qL )

T W = CS + πKQ =

Notice that wherever consumer surplus is decreasing in either K or Q, the third term of the above equation is increasing, so do the first two terms which means that total welfare is always an increasing function of both capacity levels.  Proof of Proposition 4 Given Lemma 1, optimal capacities are chosen from the KQ region which the following maximization problem: (vH − vL )Q (vL qH − vH qL )K Q2 + E(q 2 )K 2 − 2KQE(q) max π = + − − cK − dQ K,Q q H − qL q H − qL α(1 − α)(qH − qL )2 Q/qH ≤ K

(λ1 )

K ≤ Q/qL

(λ2 )

g(Q) ≤K ≤ h(Q) K≥0, Q≥0 ∂π

KQ Firstly, consider the interior solution where λ1 = λ2 = 0.From ∂K = c and ∂πKQ = d one immediately gets the optimal capacity levels described in part 1 ∂Q of Proposition 4. Replacing these optimal values into conditions Q/qH ≤ K and K ≤ Q/qL imply vH > c + dqH and vL > c + dqL , respectively. The remaining four primal feasibility conditions are satisfied at this solution.

Secondly, consider λ1 > 0 and λ2 = 0 that corresponds to the exclusion of low-types. K = Q/qH and the two first order conditions provide the optimal capacity levels described in part 2 of Proposition 4. Positivity of λ1 requires vL ≤ c + dqL , and the positivity of Q implies vH > c + dqH .The remaining four primal feasibility conditions are satisfied. Thirdly, consider λ2 > 0 and λ1 = 0 that corresponds to the exclusion of high-types. K = Q/qL and the two first order conditions provide the optimal capacity levels described in part 3 of Proposition 4. Positivity of λ2 requires vH ≤ c + dqH , in addition, the positivity of K implies vL > c + dqL . The remaining four primal feasibility conditions are satisfied. 37

s.t.

Both capacity levels are zero if the marginal cost of serving each consumer group is prohibitively high, which corresponds to part 4 of Proposition 4. Finally, replacing the optimal capacity levels into the optimal prices in region KQ, one immediately gets that pi = (vi + c + dqi )/2, i ∈ {L, H}.  Proof of Proposition 5 Showing that the outcome of (P-ICL) coincides with the outcome of (P-LH) consists of showing that in each local optimum, pL ≤ pH is satisfied. Optimal prices in regions K, KQ and U trivially satisfy this condition. In region Q, it is equivalent to

vL −

KqH − Q Q − KqL 1 ≤ vH − ⇔ K≥ (Q − α(1 − α)(qH − qL )(vH − vL )) , (1 − α)(qH − qL ) α(qH − qL ) E(q)

which by the definition of g(Q) is equivalent to K ≥ g(Q) −

1 vH − vL α(1 − α)(qH − qL ) . E(q) 2

This condition is always satisfied in region Q, since in this region the stronger condition of K ≥ g(Q) is also satisfied. Next, I show that the solutions of (P-ICL2) are always dominated by some solution of (P-ICL). Notice that pH = E(v)/2 is only attainable if K > E(v)/2 and Q > E(v)qH /2 meaning that the capacities are from the U region, where the solution of (P-ICL2) is clearly dominated by π U . The two conditions can be rewritten as E(v) − min(K, Q/qH ) ≤ pH < vL , which immediately implies that a necessary condition for existence of a solution is min(K, Q/qH ) > α(vh − vL ). This in turn implies that the solution cannot be in the EL region. By concavity of the objective function, whenever solution exists, it is given by pH = E(v) − min(K, Q/qH ) thus the profit equals E(v)K − K 2 . For K < Q/qH the necessary condition implies that capacity levels fall in the K region, where πK is attainable in (P-ICL). By definition,  πK = α(1 − α)

vH − vL 2

2

+ KE(v) − K 2 > KE(v) − K 2 ,

so if K < Q/qH , the optimal solution is dominated. Next, I show that the same 38

is true for K ≥ Q/qH . In this case, for all K we have KE(v) − K 2 > E(v)Q/qH − (Q/qH )2 , where the right-hand side is the optimal profit of (P-ICL2). The left-hand side is smaller than πK , as shown above, therefore it is also dominated by the optimal profits attainable in regions KQ and Q in the (P-ICL) problem. Proof of Lemma 2 i (p) (i) Myerson’s regularity condition, i.e., p − 1−F , i ∈ {L, H} being strictly fi (p) increasing in p can be rewritten in terms of the demand function as

Di00 (p) <

2(Di0 (p))2 . Di (p)

Firstly, consider the slope of curve h(Q), defined as the part of the KQ’ region where λ2 = 0. Substituting λ2 = 0 in the first order conditions, one gets

pH +

DH (pH ) DL (pL ) = pL + 0 0 DH (pH ) DL (pL )

⇐⇒

0 0 ξ(K, Q) ≡ DL DH −DH DL0 +(pL −pH )DL0 DH =0

∂ξ/∂Q The implicit function theorem ensures that h0 (Q) = ∂K = − ∂ξ/∂K . In the ∂Q following I show that this derivative is positive if the regularity condition is satisfied. ∂x , i.e., for any function x, x denotes its partial derivative with respect to Let x = ∂Q KqH −Q Q−KqL Q. We know that DL (pL ) = (1−α)(q and DH (pH ) = α(q , which implies H −qL ) H −qL )

DL =

−1 (1 − α)(qH − qL )

and DH =

1 α(qH − qL )

and pi =

Di Di0

and Di0 = Di00

Di Di0

for i ∈ {L, H}. We have ∂ξ ∂DL ξ= = ∂Q ∂Q



0 2DH

+ ((pL −

0 pH )DH

   00 DL00 ∂DH DH 0 0 − DH ) 0 + 2DL − ((pL − pH )DL + DL ) 0 DL ∂Q DH

Similarly, ∂ξ ∂DL = ∂K ∂K

    00 DL00 DH ∂DH 0 0 0 0 2DH + ((pL − pH )DH − DH ) 0 + 2DL − ((pL − pH )DL + DL ) 0 . DL ∂K DH

From the implicit function theorem:

39

−A − ∂K (1−α)(qH −qL ) =− AqH ∂Q + (1−α)(qH −qL )

B α(qH −qL ) BqL α(qH −qL )

=

αA + (1 − α)B αAqH + (1 − α)BqL

where

0 0 A = 2DH + ((pL − pH )DH − DH )

DL00 DL0

and B = 2DL0 − ((pL − pH )DL0 + DL )

00 DH 0 DH

> 0 is that both A and B be negative. Next, I show A sufficient condition for ∂K ∂Q that the regularity condition for FL and FH are equivalent to A < 0 and B < 0, respectively.

A<0

⇐⇒

0 0 DL00 (DH −(pL −pH )DH ) < 2DL0 DH

⇐⇒

DL00 <

0 2DL0 DH 0 0 DH + pH DH − pL D H

The first inequality comes from DL0 < 0. The second inequality’s direction follows 0 > 0 which from the positive slope of the profit curve that guarantees DH + pH DH 0 0 0 in turn implies that DH + pH DH − pL DH > 0. Simplifying the ratio by DH leads to

DL00

2DL0 < 0 DH /DH + pH − pL

⇐⇒

DL00

2DL0 < DL /DL0 + pL − pL

⇐⇒

DL00

2(DL0 )2 < DL

where the first inequality comes from the first order condition of DH L pH + D = pL + D The last inequality corresponds exactly to the regular0 0 . DL H ity of FL . Similar arguments prove that B < 0 is equivalent to the regularity of FH , which concludes the proof of h0 (Q) > 0. Analogous arguments can be made with obvious modifications that show that g is also increasing under the regularity assumption.  (ii) By definition K is slack for K > h(Q) and Q is slack for K < g(Q). By b < Q. Then by continuity there contradiction, assume that h < g at some point Q b where h < g. For any Q in this interval and for any exists a neighborhood of Q K < h(Q) we have none of the constraints binding. However, this is impossible, as the unconstrained optimum can only be achieved for (Q, K) ≥ (Q, K). Moreover, g(Q) = h(Q) = K follows directly from the definition of Q and K.  (iii) By contradiction, assume that ∀ Q > 0 : g(Q) > Q/qH . This means that ∀ Q > 0 : ∃ (Q) > 0 such that g(Q) > Q/qH + (Q). By definition of g, the

40

constraint on Q is slack for all capacity-pairs (Q, Q/qH + (Q). The constraint on Q being slack translates to qH DH (pH ) + (1 − α)qL DL (pL ) < Q and the constraint on K at point (Q, Q/qH + (Q) writes as αDH (pH ) + (1 − α)DL (pL ) ≤ K = Q/qH + (Q) Taking the limit of Q → 0, the first inequality implies DL and DH must also tend to 0. Thus the second constraint must also be slack as the left hand side tends to zero while the right hand side equals (Q) which is always strictly positive. However, this is a contradiction as both constraints cannot be slack unless they are larger then (Q, K). (iv) can be proven with arguments analogous to (iii).  Proof of Proposition 6 It is sufficient to show that the optimal solutions of (P-LH) satisfy the additional constraint of pH /qH ≤ pL /qL . In the K region, this condition writes as     1 1 vH − vL vH − vL −K ≤ −K , vH − (1 − α) vL + α qH 2 qL 2 which can be rewritten as K<

vL qH − vH qL E(q)(vH − vL ) . + qH − qL 2(qH − qL )

Elementary algebra shows that the expression on the right-hand side is greater than E(v)/2, so the condition is always satisfied in the K region. In the KQ region, we have

pH /qH ≤ pL /qL ⇔ K < f (Q) +

1 α(1 − α)(vL qH − vH qL )(qH − qL ), 2E(q 2 )

which is always satisfied as the KQ region is delimited by K ≤ f (Q) and the right-hand side is greater than f (Q). It is straightforward to see that the constraint is also satisfied in regions Q and U.  Proof of Proposition 7 As the consumer surplus is additive in the consumer surplus of the consumer groups, we have

41

∂CS ∂pH ∂ = ∂K ∂K ∂pH

Z

θH

pH

∂pL ∂ α(w − pH )fH (w)dw + ∂K ∂pL

Z

θL

(1 − α)(w − pL )fL (w)dw. pL

Using the Leibniz-rule it follows that qH −qL ∂CS = (−(1 − α)DL ) + (−αDH ) 0 0 ∂K (1 − α)(qH − qL )DL α(qH − qL )DH which implies that ∂CS <0 ∂K

⇐⇒

qL

DL0 (pL ) D0 (pH ) < qH H . DL (pL ) DH (pH )

Analogous steps prove the second statement. 

References Acemoglu, D., K. Bimpikis, and A. Ozdaglar (2009): “Price and capacity competition,” Games and Economic Behavior, 66(1), 1 – 26. Adan, I., and J. Vissers (2002): “Patient mix optimisation in hospital admission planning: a case study,” International Journal of Operations & Production Management, 22(4), 445–461. Aguirre, I., S. Cowan, and J. Vickers (2010): “Monopoly price discrimination and demand curvature,” The American Economic Review, 100(4), 1601–1615. Banditori, C., P. Cappanera, and F. Visintin (2013): “A combined optimization-simulation approach to the master surgical scheduling problem,” IMA Journal of Management Mathematics. Bergemann, D., B. A. Brooks, and S. Morris (forthcoming): “The limits of price discrimination,” The American Economic Review. Bertsimas, D., and R. Shioda (2003): “Restaurant revenue management,” Operations Research, 51(3), 472–486. Besanko, D., and R. Braeutigam (2010): Microeconomics. John Wiley & Sons. Bulow, J., and J. Roberts (1989): “The Simple Economics of Optimal Auctions,” Journal of Political Economy, 97(5), 1060–1090.

42

Cowan, S. (2007): “The welfare effects of third-degree price discrimination with nonlinear demand functions,” RAND Journal of Economics, 38(2), 419–428. (2012): “Third-Degree Price Discrimination and Consumer Surplus,” The Journal of Industrial Economics, 60(2), 333–345. Cripps, M., and N. Ireland (1988): “Equilibrium and capacities in a market of fixed size,” Manuscript, University of Warwick. Davidson, C., and R. Deneckere (1986): “Long-Run Competition in Capacity, Short-Run Competition in Price, and the Cournot Model,” The RAND Journal of Economics, 17(3), 404–415. de Frutos, M.-Á., and N. Fabra (2011): “Endogenous capacities and price competition: The role of demand uncertainty,” International Journal of Industrial Organization, 29(4), 399–411. Kasilingam, R. G. (1997): “Air cargo revenue management: Characteristics and complexities,” European Journal of Operational Research, 96(1), 36–44. Kimes, S. E., and G. M. Thompson (2004): “Restaurant revenue management at Chevys: Determining the best table mix,” Decision Sciences, 35(3), 371–392. Kreps, D. M., and J. A. Scheinkman (1983): “Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes,” The Bell Journal of Economics, 14(2), 326–337. Reynolds, S. S., and B. J. Wilson (2000): “Bertrand-Edgeworth Competition, Demand Uncertainty, and Asymmetric Outcomes,” Journal of Economic Theory, 92(1), 122–141. Xiao, B., and W. Yang (2010): “A revenue management model for products with two capacity dimensions,” European Journal of Operational Research, 205(2), 412– 421.

43

Monopoly pricing with dual capacity constraints

Sep 14, 2015 - Email address: [email protected] The latest version ...... 6One can see the case of very large Q as an alternative benchmark.

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